Without Mapquesting it, all are roughly the same distance from Oxford, (and all coincidentaly require driving past Sunday River Brewing
). I would guess Baldpates via Frye Brook to be the farthest away, then Speck (or Baldpate via the AT), Andover White Cap, then Puzzle, then Goose Eye. Having not done Puzzle I cant comment, but in terms of difficulty (and time on trail), It would be Speck and Goose Eye comparible and Baldpates either direction next and White cap last
The bonus spots for each trip are
Speck - Eyebrow cliff and the view from the summit tower (used to be a viewless peak until they rebuilt the tower and cut in the Helipad). Good excuse to stop by and see Screw Auger Falls.
Goose Eye - Nice loop hike into a undisturbed cirque. Great views along the ridge, plenty of places to stop for a swim along the brook. Easy to spend a long day there.
Baldpates - Good views into the notch, the approach via Frye Brook as lots of water and Frye Brook has a lot of interesting slot canyons and water falls.
White Cap - Good views from a diferent perspective. Potentially more blue berries than someone could eat in a year. A short day for those who want to get back to Oxford and Party.
Note that the Elephant crew is on the road past Andover White Cap.
As for me if the weather is hot and muggy, probably Baldpates via Frye Brook, if sunny and breezy, White Cap. I will wait until late week to make the call.